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Almagest Book IV: The Babylonian Eclipse Triple – Solar/Lunar Positions & Epicyclic Motion of Anomaly

So far in this book we’ve covered the ancient Greek values for the various motions of the moon. For the most part Ptolemy has accepted them as authoritative, but to demonstrate some of the methodology, Ptolemy wants to walk us through one: The lunar anomaly.

We shall use, first, among the most ancient eclipses available to us, three [which we have selected] as being recorded in an unambiguous fashion, and, secondly, [we shall repeat the procedure] using among contemporary eclipses, three which we ourselves have observed very accurately. In this way our results will be valid over as long a period as possible, and in particular, it will be apparent that approximately the same equation of anomaly results from both demonstrations, and that the increment in the mean motion [between two sets of eclipses] agrees with that computed from the above periods.

Continue reading “Almagest Book IV: The Babylonian Eclipse Triple – Solar/Lunar Positions & Epicyclic Motion of Anomaly”

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Almagest Book IV: Uneven Ratios in Epicyclic and Eccentric Model

In the last post, we explored how the eccentric and epicyclic models could produce the same result even if you didn’t have the same period of anomaly and ecliptic longitude. This was done by allowing the center of the eccentre to rotate around the Earth. In this post, we’ll explore how they can still produce the same result even

if [the members of] the ratios are unequal, and the eccentre is not the same size as the deferent…provided the ratios are similar Continue reading “Almagest Book IV: Uneven Ratios in Epicyclic and Eccentric Model”

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Almagest Book IV: In the Simple Hypothesis of the Moon, the Same Phenomena are Produced by Both the Eccentric and Epicyclic Hypotheses

Our next task is to demonstrate the type and size of the moon’s anomaly.

In chapter 2 of this book, we spent quite a bit of time talking about the moon’s anomaly, describing a method by which Hipparchus could have used periods of eclipses to determine the anomaly’s period. While we never actually completed the method, Ptolemy still gave us the period Hipparchus supposedly derived. Now we’re going to put that to use to start building our first lunar model. Continue reading “Almagest Book IV: In the Simple Hypothesis of the Moon, the Same Phenomena are Produced by Both the Eccentric and Epicyclic Hypotheses”

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Almagest Book IV: Lunar Mean Motion Tables

The fourth chapter of Book IV takes what we worked on in the last post and expands it for convenient reference. As with the solar mean motion tables we created back in Book III, Ptolemy lays this one out in several intervals: 18 year periods, single year periods, months, days, and hours.

These tables essentially answer the question: “If the moon’s mean position was as X, if I waited Y interval of time, where would it be then?” Continue reading “Almagest Book IV: Lunar Mean Motion Tables”

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Almagest Book IV: Favorable Positions for Lunar Eclipse Pairs

Now that we’ve covered the positions the sun needs to be in to avoid its anomaly influencing things, and the positions to avoid for the moon, so its anomaly doesn’t influence things, we’ll look into some positions which would make it the most obvious if the above were. Ptolemy states this saying,

we should select intervals [the ends of which are situated] so as to best indicate [whether the interval is or is not a period of anomaly] by displaying the discrepancy [between two intervals] when they do not contain an integer number of returns in anomaly.

So which are those? Continue reading “Almagest Book IV: Favorable Positions for Lunar Eclipse Pairs”

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Almagest Book IV: The Lunar Anomaly and Eclipses

In the last post, we covered how the sun’s anomaly impacts things, but

we must pay no less attention to the moon’s [varying] speed. For if this is not taken into account, it will be possible for the moon, in many situations, to cover equal arcs in longitude in equal times which do not at all represent a return in lunar anomaly as well.

I’ll preface this section by saying this is, to date, by far the hardest section I’ve grappled with. I believe a large part of the difficulty came from the fact that Ptolemy is exceptionally unclear about what his goal is with this section. My initial belief was that it was to find the full period in which a the position of the sun and moon would “reset” as discussed in the last post. However, that’s something we’re going to have to work up to.

For now, we’re going to concentrate on just one of the various types of months. Namely, the “return in lunar anomaly” which is another way of saying the anomalistic month. Continue reading “Almagest Book IV: The Lunar Anomaly and Eclipses”

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Data: Stellar Quadrant Observations – 4/21/20

I managed to get out and do a bit of observing last night. It’s been 4 months to do the day since my last opportunity to do so and I started off by making a rather large blunder. The location I usually observe at has several pads for observing and I have one I usually use. It’s straight down a path that points generally north which makes finding the north star fairly easy, even without looking for the big dipper to verify which star it is.

Well, last night I looked for the star over where the path met the parking lot, but because I was on a pad a bit to the west of where I normally am, I accidentally aligned on Kochab, the second brightest star in Ursa Minor, and ended up making three observations over the course of a half hour before I realized the error. Fortunately, we still got in 11 observations, and all of them were new stars! Continue reading “Data: Stellar Quadrant Observations – 4/21/20”

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Almagest Book IV: The Solar Anomaly and Lunar Periods

In the last post, we explored various lunar cycles from astronomers predating Ptolemy in which the moon reset its ecliptic longitude and anomalistic motion to define a full lunar period. These ancient astronomers did this by studying pairs of lunar eclipses1but Ptolemy notes that this method

is not simple or easy to carry out, but demands a great deal of extraordinary care

The reason for this difficulty is that, without careful consideration there can essentially be false positives of eclipses separated equally in time, but do not in fact, result in the moon returning to the same ecliptic longitude or same speed.

One of the reasons is that the conditions necessary to produce a lunar eclipse are also dependent on the sun, which has anomalistic motion. As such, it could be entirely possible that the moon could not have yet returned to the same ecliptic longitude as a previous eclipse, but the sun’s anomaly could cause an eclipse anyway.  Thus, a pair of eclipses may be equally separated in time, but

this is no use to us unless the sun too exhibits no effect due to anomaly, or exhibits the same [anomaly] over both intervals: for if this is not the case, but instead, as I have said, the equation of anomaly has some effect, the sun will not have travelled equal distances over [the two] equal time intervals, nor, obviously, will the moon.

To illustrate this, Ptolemy starts with an example. Continue reading “Almagest Book IV: The Solar Anomaly and Lunar Periods”